A variant on a classical question about eigenvalues of sums of Hermitian matrices is the following: What are the possible eigenvalues for a collection of unitary matrices whose product is the identity? I will explain how a fundamental inequality (proved by Falbel and Wentworth) for the logarithms of eigenvalues of unitary matrices can be used to define a twisted pullback map from vector bundles on a smooth Deligne-Mumford quotient stack ${\mathcal X}$ to vector bundles on the double inertia stack $I_{\mathcal X} \times_{\mathcal X} I_{\mathcal X}$. The twisted pullback map can then be used to define orbifold products on the cohomology/Chow ring/$K$-theory of the inertia stack $I_{\mathcal X}$. This construction, which completes a program begun by Fantechi-G\"ottsche and Jarvis-Kaufmann-Kimura, gives a method for defining orbifold products without reference to moduli spaces of twisted stable maps. This talk is based on joint work with Tyler Jarvis and Takashi Kimura.